Chapter 1.5, Problem 1E

Analyze the logical form of each of the following statements, and construct just the outline of a proof by the given method. Do not provide any details of the proof.

1. Outline a proof by contraposition that if $\left(G,*\right)$ is a cyclic group, then $\left(G,*\right)$ is abelian.
2. Outline a proof by contraposition that if B is a nonsingular matrix, then the determinant of B is not zero.
3. Outline a proof by contradiction that the set of natural numbers is not finite.
4. Outline a proof by contradiction that if x is a nonzero real number, the multiplicative inverse of x is unique.
5. Outline a two-part proof that the inverse of the function f from A to B is a function from B to A if and only if f is one-to-one and fis onto B.
6. Outline a two-part proof that a subset A of the real numbers is compact if and only if A is closed and bounded.